The above relations show that finite cyclic groups play important roles in the study of Fuglede's conjecture in \(\mathbb {R}\) .
As we have seen, Fuglede's conjecture holds in the following finite cyclic groups: \(\mathbb {Z}_{p^{n}}\)  [1]}, \(\mathbb {Z}_{p^{n}q^{m}}\)  (\(p<q\)  and \(m\le 9\)  or \(n\le 6\) ; \(p^{m-2}<q^{4}\) ) [2]}, [3]}, [4]}, \(\mathbb {Z}_{pqr}\)  [5]}, \(\mathbb {Z}_{p^{2}qr}\)  [6]} and \(\mathbb {Z}_{pqrs}\)  [7]}, where \(p,q,r,s\)  are distinct primes. For the direction “Tile \(\Rightarrow \)  Spectral”, Łaba [1]} proved \(T-S(\mathbb {Z}_{p^nq^m})\)  for distinct primes \(p,q\) . Later, Łaba and Meyerowitz proved \(T-S(\mathbb {Z}_{n})\)  in comments of Tao's blog [9]} (see also [5]}), where \(n\)  is a squarefree integer. Recently, Malikiosis [3]} proved \(T-S(\mathbb {Z}_{p_{1}^{n}p_2\cdots p_k})\) , where \(p_1,p_2,\dots ,p_k\)  are distinct primes. In [12]}, [13]}, the authors developed some new tools to study tiling sets in cyclic groups and proved \(T-S(\mathbb {Z}_{p^2q^2r^2})\) , where \(p,q,r\)  are distinct primes.
